Additivity of crossing number

A classical conjecture in knot theory says that when you have a connect-sum of knots, you obtain a minimal-crossing planar diagram for it by taking the minimal-crossing planar diagrams of the prime summands, and take the connect-sum of their diagrams. Said another way "crossing number of knots is additive". That got me to wondering, are there similar possibilities for other operations on knots and links, related to crossing number?

The above photo is a planar diagram for a 'satellite knot'. There's 24crossings in the diagram. If additivity of crossing numbers were to generalize to JSJ-decompositions, I suspect there should be no diagrams for this link with less than 24 crossings. What do you think?